A Simple Converse of Burnashev's Reliability Function

In a remarkable paper published in 1976, Burna- shev determined the reliability function of variable-length block codes over discrete memoryless channels (DMCs) with feedback. Subsequently, an alternative achievability proof was obtained by Yamamoto and Itoh via a particularly simple and instructive scheme. Their idea is to alternate between a communication and a conﬁrmation phase until the receiver detects the codeword used by the sender to acknowledge that the message is correct. We provide a converse that parallels the Yamamoto–Itoh achievability construction. Besides being simpler than the original, the pro-posed converse suggests that a communication and a conﬁrmation phase are implicit in any scheme for which the probability of error decreases with the largest possible exponent. The proposed converse also makes it intuitively clear why the terms that appear in Burnashev’s exponent are necessary. Index Terms—Burnashev’s error exponent, discrete memoryless channels (DMCs), feedback, reliability function, variable-length communication.